Emission Properties of Periodic Fast Radio Bursts from the Motion of Magnetars: Testing Dynamical Models
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Emission Properties of Periodic Fast Radio Bursts from the Motion of Magnetars: Testing Dynamical Models
Dongzi Li
1, 2, 3 and J. J. Zanazzi Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario, M5S 1A7, Canada Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada Cahill Center for Astronomy and Astrophysics, California Institute of Technology, 1216 E California Blvd, Pasadena, CA 91125, US
Submitted to AJABSTRACTRecent observations of the periodic Fast Radio Burst source 180916.J0158+65 (FRB 180916) find small linearpolarization position angle swings during and between bursts, with a burst activity window that becomes bothnarrower and earlier at higher frequencies. Although the observed chromatic activity window disfavors modelsof periodicity in FRB 180916 driven by the occultation of a neutron star by the optically-thick wind from astellar companion, the connection to theories where periodicity arises from the motion of a bursting magnetarremains unclear. In this paper, we show how altitude-dependent radio emission from magnetar curvatureradiation, with bursts emitted from regions which are asymmetric with respect to the magnetic dipole axis,can lead to burst activity windows and polarization consistent with the recent observations. In particular, thefact that bursts arrive systematically earlier at higher frequencies disfavors theories where the FRB periodicityarises from forced precession of a magnetar by a companion or fallback disk, but is consistent with theorieswhere periodicity originates from a slowly-rotating or freely-precessing magnetar. Several observational testsare proposed to verify/differentiate between the remaining theories, and pin-down which theory explains theperiodicity in FRB 180916.
Keywords: radiation mechanisms: general – polarization – stars: neutron – stars: magnetars INTRODUCTIONFast radio bursts (FRBs) are short ( ∼ 𝜇 s to ∼ ms duration)radio bursts with unknown origins. A decade after the firstdiscovery (Lorimer et al. 2007), massive progress has beenmade in understanding the nature of FRBs. The discoveryrepeating FRBs (e.g. Spitler et al. 2014, 2016; CHIME/FRBCollaboration et al. 2019a; Fonseca et al. 2020) suggests atleast a portion of the FRB population has a non-catastrophicorigin. The first localization of an FRB source (Chatterjeeet al. 2017) confirmed their cosmological origin. The detec-tion of extremely bright radio bursts from a galactic magnetar(CHIME/FRB Collaboration et al. 2020b; Bochenek et al.2020) places magnetars as a most promising progenitor forFRBs.Repeating FRBs offer great opportunities for follow-upobservations, as well as studying the burst properties withtime and frequency. FRB 180916.J0158+65 (hereafter FRB Corresponding author: Dongzi [email protected] 𝑧 = . HubbleSpace Telescope shows that the location of FRB 180916 is off-set by ∼
250 pc from the nearest knot of star formation in the a r X i v : . [ a s t r o - ph . H E ] J a n Li & Zanazzihost galaxy (Tendulkar et al. 2020) , while young magnetarsare expected to born in a star forming region. For bursts de-tected in the 𝐿 -band, the change of linear polarization positionangle (PA) is constrained to be (cid:46) ◦ -20 ◦ across the burstsof the same phase (Nimmo et al. 2020), and (cid:46) ◦ acrossthe 𝐿 -band active phases (Pastor-Marazuela et al. 2020). NoPA swing is observed, as would generically be expected inthe magnetospheric origin models (as oppose to the diversepolarization angle swing detected in FRB 180301, Luo et al.2020). Most strikingly, the active phase is observed to bechromatic, with the activity window being both narrower andearlier at higher frequencies (Pastor-Marazuela et al. 2020;Pleunis et al. 2020). This disfavours models explaining theperiodic activity with the eclipse of a companion wind, asthese theories predict a narrower activity window at lowerfrequencies (Lyutikov et al. 2020; Ioka & Zhang 2020).Recently, a period of ∼
160 days was detected in the re-peating FRB 121102 (Rajwade et al. 2020; Cruces et al.2021). Similar to FRB 180916, FRB 121102 also has smallPA swings during and between bursts (Michilli et al. 2018),but data on the frequency-dependence of the activity windowhas yet to be gathered. Clearly, the polarization emissionand frequency-dependence expected from different models ofperiodic FRBs is becoming highly topical.The goal of this work is to show how the recent constraintson the PA variation, as well as the dependence on the FRBactivity window with frequency, fit into theories which arguethe periodicity of FRB 180916 originates from the motionof a Neutron Star (NS) or magnetar. Section 2 justifies ourphenomenological emission model for NSs emitting FRBs.Section 3 discusses the three dynamical theories which havebeen put forth to explain the periodicity of FRB 180916,which we test in this work: a NS with a long rotation period,a NS undergoing free precession, and a NS undergoing forcedprecession. Section 4 presents the main results of this work,and discusses which dynamical models are favored to explainthe periodicity of FRB 180916. Section 5 discusses futureobservations which can potentially distinguish between theremaining theories. Section 6 summarizes our work. EMISSION MODEL FOR FRBSTo make testable predictions for periodic FRB emission, weconstruct a phenominological model of radio emission froma rotating (and possibly precessing) NS, motivated by modelsfor radio pulsar emission (e.g. Ruderman & Sutherland 1975;Rankin 1993). Figure 1 displays the geometry of the model.A NS with rotation axis ˆ 𝝎 and dipole moment unit vector ˆ 𝒑 isviewed by an observer, with ˆ 𝒏 being the unit vector pointingin the direction of the Line of Sight (LOS) of the observer.FRBs are emitted in a direction ˆ 𝒎 𝑓 with frequency 𝑓 , and anobserver sees the FRB emission when ˆ 𝒏 lies within the conecentered on ˆ 𝒎 𝑓 with opening angle 𝜉 𝑓 (orange region of Figure 1.
Geometry of our model for FRB emission in the upper-half of a NS’s magnetosphere. The FRB emission at frequency 𝑓 originates from a cone centered around ˆ 𝒎 𝑓 , with opening angle 𝜉 𝑓 .See §2 for details. Fig. 1). The emission axis ˆ 𝒎 𝑓 is offset from ˆ 𝒑 by a magneticpolar angle 𝛿 ob (angle between ˆ 𝒎 𝑓 and ˆ 𝒑 ), and a magneticlongitude 𝛼 (angle between planes spanned by vector pairs { ˆ 𝒙 , ˆ 𝒑 } and { ˆ 𝒎 𝑓 , ˆ 𝒑 } ). The magnetic (angle between ˆ 𝒑 and ˆ 𝝎 ) and emission (angle between ˆ 𝒎 𝑓 and ˆ 𝝎 ) angles are 𝜒 and 𝜒 𝑓 , respectively, while the angle between ˆ 𝒏 and ˆ 𝝎 is 𝜈 . Theangles between ˆ 𝒑 , ˆ 𝝎 , and ˆ 𝒏 , and the symmetry axis ˆ 𝒛 , are 𝜃 𝑝 , 𝜃 𝜔 , and 𝜃 𝑛 , respectively. The position angle PA of the linearpolarization (in the rotating vector model, Radhakrishnan &Cooke 1969) is the angle between the planes spanned by thevector pairs { ˆ 𝒏 , ˆ 𝝎 } and { ˆ 𝒏 , ˆ 𝒑 } .The emission model assumed in this paper is significantlymore complex than the model in Zanazzi & Lai (2020), whichassumed ˆ 𝒎 𝑓 = ˆ 𝒑 . In this section, we justify our model as-sumptions about the FRB emission region, assuming the co-herent radio emission arises due to curvature radiation fromcharged particles travelling along the NS magnetic field lines.Magnetar curvature radiation has already been invoked bymany models to explain the coherent radio emission char-acteristic of FRBs (e.g. Katz 2014; Cordes & Wasserman2016; Kumar et al. 2017; Lu & Kumar 2018; Yang & Zhang2018; Lu et al. 2020; Yang et al. 2020) Section 2.1 discussesour expectations for how the emission direction ˆ 𝒎 𝑓 changeswith FRB frequency 𝑓 , while Section 2.2 discusses the linearpolarization of the emission.2.1. FRB Emission Direction
In the study of pulsar emission, it is widely believed thatdifferent frequencies are produced at different altitudes withinthe pulsar magnetosphere (e.g. Manchester & Taylor 1977).It has been shown the spread of field lines at higher altitudecould account for the widening of pulse profiles at lowereriodic FRB Emission from Magnetar Motion 3 r sin δ ob /R r c o s δ o b / R δ em δ ob r field lines0 20 40 60 80 100 δ ob (degrees) N u m b e r o f bu r s t s Figure 2.
Altitude and angular dependence of FRB emissionfrom curvature radiation (eq. [5]), for 55 ≤ 𝐶 ≤ Top panel:
Trajectories of magnetic field lines (eq. [1]). Field lines whichcause emission in different frequency bands are colored green (1.36-2 GHz), orange (400-800 MHz), and blue (110-180 MHz). Themagnetic polar angles 𝛿 em and 𝛿 ob are defined by the black dashedlines. Bottom panel:
Histogram of observed magnetic polar angle 𝛿 ob of bursts from a centered dipole magnetic field. The histogramis computed assuming an equal number of bursts are emitted perunit length of the field line, with 𝐶 spaced linearly, for the frequencybands 1.36-2 GHz (green), 400-800 MHz (orange), and 110-180MHz (blue). The normalization for the Number of Bursts is arbitrary.The altitude of the burst emission, as well as the mean and varianceof the burst emission 𝛿 ob values, increase with decreasing burstfrequency 𝑓 . frequencies (e.g. Cordes 1978; Phillips 1992). This kind of“radius-to-frequency mapping” has also been used to explainthe downward drifting pattern seen in some repeating FRBs(Wang et al. 2019; Lyutikov 2020). Here, we demonstrate thissame effect in the context of curvature radiation, and showhow it will lead to the magnetic polar angle 𝛿 ob and coneopening angle 𝜉 𝑓 of the observed FRB emission to decreasewith an increase in frequency 𝑓 .For simplicity, we assume the magnetic field exterior to theNS is a dipole in vacuum, and ignore how plasma affects themagnetic field (e.g. Goldreich & Julian 1969; Tchekhovskoy NS EmissionRegion
Figure 3.
Cartoon representation of offset magnetic field geometry,which can give rise to FRB emission from curvature radiation con-sistent with our model (Fig. 1). The strong exposed magnetic fieldson one side of the NS lead to burst emission occuring in “cones”around the emission axis ˆ 𝒎 𝑓 , with both the magnetic polar angle 𝛿 ob (angle between ˆ 𝒎 𝑓 and ˆ 𝒑 ) and opening angle 𝜉 𝑓 increasingwith decreasing frequency 𝑓 . et al. 2016; Philippov & Spitkovsky 2018). The trajectory ofa single field line in polar coordinates is given by 𝑟𝑅 = 𝐶 sin 𝛿 em (1)where 𝛿 em is the magnetic polar angle of the field line, 𝑟 is thedistance to the center of the NS, 𝑅 is the NS radius, while1 ≤ 𝐶 < ∞ is a constant which varies for different field lines.Emission at 𝛿 em will be observed by a distant observer at adifferent magnetic polar angle 𝛿 ob , which is related to 𝛿 em bycos 𝛿 ob = + 𝛿 em √ + 𝛿 em . (2)The trajectory of a number of different magnetic field lines,as well as the relation between 𝛿 em and 𝛿 ob , are displayed inthe top panel of Figure 2.The characteristic frequency 𝑓 of emission from curvatureradiation is (e.g. Ruderman & Sutherland 1975): 𝑓 = 𝜋 𝛾 𝑐𝜌 , (3)where 𝛾 is the Lorentz factor, while 𝜌 is the curvature radius.For a dipolar field (eq. [1]), 𝜌 can be shown to be (e.g. Yang& Zhang 2018; Wang et al. 2019) 𝜌 = 𝑟 ( + 𝛿 em ) 𝛿 em ( + cos 𝛿 em ) ≡ 𝑟 𝐹 ( 𝛿 em ) . (4)Assuming for our simple model, the Lorentz factor decreaseswith 𝑟 as 𝛾 ( 𝑟 ) = 𝛾 ( 𝑟 / 𝑅 ) − / (also assumed in e.g. Lyutikov2020 to explain the downward drifting rate of FRBs), theemission frequency becomes 𝑓 = 𝐾 𝐹 ( 𝛿 em ) (cid:18) 𝑟𝑅 (cid:19) − (5) Li & Zanazziwhere 𝐾 = 𝑐𝛾 /( 𝜋𝑅 ) , with 𝑅 = cm and 𝐾 = 𝛾 ≈ .The upper panel of Figure 2 shows the emission heights ofdifferent frequency bands, displayed in green (1 . − −
800 MHz), and blue (110 −
180 MHz). Clearlyfor curvature radiation, high (low) frequency radiation orig-inates at low (high) altitudes. The bottom panel displays ahistogram for number of bursts with a given 𝛿 ob value, as-suming an equal number of bursts are emitted per unit lengthof the field line, with the field line 𝐶 values spaced linearlybetween 55 ≤ 𝐶 ≤
80. From this, we see both the meanand spread of 𝛿 ob values for a given frequency band increaseswith decreasing frequency.Within this simple model, a dipole situated at the centerof a NS will lead to FRB emission symmetric about the NSdipole moment ˆ 𝒑 , rather than localized on a small “patch”at a location displaced from ˆ 𝒑 . One simple modification tothe magnetic field geometry which can lead to asymmetricemission about the ˆ 𝒑 axis is displayed in Figure 3: a dipoledisplaced from the NS center. The strong magnetic fields ex-posed on one side of the NS (which are buried on the other sidein this cartoon) can lead to emission “cones” similar to thoseassumed in our model (Fig. 1), with opening angles compa-rable to the width of the histogram widths displayed in thebottom panel of Figure 2. There is growing observational ev-idence many NSs may have similar magnetic field geometriesas displayed in Figure 3. An offset dipole has been invoked toexplain X-ray emission from the mode-switching pulsar PSRB0943+10 (Storch et al. 2014). Recently, X-ray observationsfrom the NICER mission found the hot spots on the surfaceof isolated pulsars to be far from antipodal, implying a highlycomplex magnetic field far from the classic assumption of acentered dipole (Riley et al. 2019; Bilous et al. 2019). Forthis work, rather than attempting to construct a complex mag-netic field which can lead to FRB emission consistent withour model, we simply leave 𝛿 ob and 𝜉 𝑓 as free parameters,with the general expectation both 𝛿 ob and 𝜉 𝑓 should increasewith decreasing 𝑓 . 2.2. FRB Polarization
To model the polarization of FRB emission, we use therotating vector model (Radhakrishnan & Cooke 1969), wherethe position angle PA of the linear polarization is given bytan PA = − sin Ψ sin 𝜒 cos 𝜒 sin 𝜈 − cos 𝜈 sin 𝜒 cos Ψ , (6)where Ψ is the rotational phase ( Ψ = ˆ 𝒏 has its closest approach to ˆ 𝒑 ), while the other quantities Notice that in Equation 5, a change from 𝑟 to 𝑟 (cid:48) can be compensated by achange of 𝐾 to 𝐾 𝑟 / 𝑟 (cid:48) to emit in the same frequency. Therefore, theemission height 𝑟 in Figure 2 can be scaled with a change of 𝛾 . are displayed in Figure 1. Although the magnetic field closeto the NS may be complex, Lu et al. (2019) argued the prop-agation of a FRB across a plasma-filled NS magnetospherecauses the electric field of the burst to “freeze” in a directionperpendicular to the magnetic field when the plasma densityis sufficiently low (and the magnetic field is approximatelydipolar), implying equation (6) should be an adequate modelfor the linear polarization from FRBs. Other works have alsoused PA measurements constrain the free-precession of NSs(Weisberg et al. 2010). We neglect how additional propaga-tion effects can cause the polarization to differ from the rotat-ing vector model (e.g. Wang et al. 2010; Beskin & Philippov2012). DYNAMICS OF PERIODIC FRB MODELSModels which ascribe the periodicity of FRB 180916 fromthe motion of a magnetar fall into three different categories.The simplest dynamical model postulated the 16.3 day peri-odicity was the rotation period of the magnetar (Beniaminiet al. 2020), which implies this magnetar must have a rotationperiod much longer than those typically observed ( ∼ ˆ 𝝎 , dipole moment axis ˆ 𝒑 , emisison direction axis ˆ 𝒎 𝑓 ,and observer LOS axis ˆ 𝒏 differ between the three different dy-namical theories. We defer a discussion of the physics whichlead to these three different classes of magnetar motions tothe references above.The dynamics of these different motions are more con-veniently analyzed in either a Cartesian coordinate sys-tem anchored into and co-rotating with the NS { ˆ 𝒙 𝑏 , ˆ 𝒚 𝑏 , ˆ 𝒛 𝑏 } (body frame), or stationary with respect to a distant observer { ˆ 𝒙 𝑖 , ˆ 𝒚 𝑖 , ˆ 𝒛 𝑖 } (inertial frame). The time evolution of a vector 𝒗 ( 𝑡 ) in the body d 𝒗 / d 𝑡 | 𝑏 or inertial d 𝒗 / d 𝑡 | 𝑖 frames are relatedvia d 𝒗 / d 𝑡 | 𝑏 + 𝝎 × 𝒗 = d 𝒗 / d 𝑡 | 𝑖 , where 𝝎 = 𝜔 ˆ 𝝎 , with 𝜔 the spinfrequency of the NS.3.1. Magnetar with Slow Rotation
The left panel of Figure 4 displays our dynamical model forperiodic FRBs due to slowly-rotating NSs. We work in thebody frame { ˆ 𝒙 𝑏 , ˆ 𝒚 𝑏 , ˆ 𝒛 𝑏 } , where ˆ 𝒑 and ˆ 𝒎 𝑓 are static, with the(here static) rotation axis defining ˆ 𝒛 𝑏 ≡ ˆ 𝝎 , with ˆ 𝒙 𝑏 lying in thedirection of the projection of ˆ 𝒑 onto the plane perpendicularto ˆ 𝒛 𝑏 . Because ˆ 𝒏 is stationary in the inertial frame, it evolveseriodic FRB Emission from Magnetar Motion 5 Slow Magnetar (body frame)
Free Precession(body frame)
Forced Precession(inertial frame)
Figure 4.
Dynamics of three different periodic FRB models. Here, ˆ 𝝎 is the spin axis, ˆ 𝒏 is a unit vector in the the LOS direction, ˆ 𝒑 is the dipolemoment unit vector, and ˆ 𝒎 𝑓 represents a unit vector pointing in the direction of the FRB emission. Left panel:
The slowly rotating magnetarmodel. Here ˆ 𝝎 and ˆ 𝒎 𝑓 are fixed in the body frame { ˆ 𝒙 𝑏 , ˆ 𝒚 𝑏 , ˆ 𝒛 𝑏 } , while ˆ 𝒏 rotates around ˆ 𝝎 . The coordinates are defined by ˆ 𝒛 𝑏 = ˆ 𝝎 , and ˆ 𝒙 𝑏 bythe projection of ˆ 𝒑 onto the plane perpendicular to ˆ 𝒛 𝑏 . Middle panel:
The free precession model. Here, the precession axis ˆ 𝒛 𝑏 , as well as ˆ 𝒑 and ˆ 𝒎 𝑓 , are fixed in the body frame { ˆ 𝒙 𝑏 , ˆ 𝒚 𝑏 , ˆ 𝒛 𝑏 } , while ˆ 𝝎 precesses around ˆ 𝒛 𝑏 , while ˆ 𝒏 rotates around ˆ 𝝎 . The coordinates are defined by ˆ 𝒛 𝑏 being the axis ˆ 𝝎 precesses about in the body frame, while ˆ 𝒙 𝑏 defined by the projection of ˆ 𝒑 onto the plane perpendicular to ˆ 𝒛 𝑏 . Right panel:
Forced precession model. Here, the precession axis ˆ 𝒛 𝑖 is fixed in the system’s inertial frame { ˆ 𝒙 𝑖 , ˆ 𝒚 𝑖 , ˆ 𝒛 𝑖 } (which we work within) instead of bodyframe { ˆ 𝒙 𝑏 , ˆ 𝒚 𝑏 , ˆ 𝒛 𝑏 } . In the inertial frame, ˆ 𝒏 is fixed, while ˆ 𝝎 precesses around the precession axis ˆ 𝒛 𝑖 , with ˆ 𝒎 𝑓 and ˆ 𝒑 rotating around ˆ 𝝎 . Thecoordinates are defined by ˆ 𝒛 𝑖 being the axis ˆ 𝝎 precesses about, while ˆ 𝒙 𝑖 defined by the projection of ˆ 𝒏 onto the plane perpendicular to ˆ 𝒛 𝑖 . as d ˆ 𝒏 / d 𝑡 | 𝑏 + 𝝎 × ˆ 𝒏 = ˆ 𝒏 ( 𝑡 ) = sin 𝜈 cos 𝜑 𝑛 ˆ 𝒙 𝑏 − sin 𝜈 sin 𝜑 𝑛 ˆ 𝒚 𝑏 + cos 𝜈 ˆ 𝒛 𝑏 , (7)with 𝜑 𝑛 ( 𝑡 ) = 𝜔𝑡 + 𝜑 𝑛 ( ) the spin phase. The spinphase is offset from the rotational phase Ψ (see eq. [6])by Ψ = 𝜑 𝑛 + 𝜋 / − Δ 𝑛 , where the offset angle Δ 𝑛 = cos − (cid:2) ( cos 𝜃 𝑝 − cos 𝜃 𝜔 cos 𝜒 ) /( sin 𝜃 𝜔 sin 𝜃 𝑝 ) (cid:3) .3.2. Magnetar undergoing Free Precession
The middle panel of Figure 4 displays our dynamical modelfor a freely-precessing NS. We work in the body frame, wherethe Cartesian coordinates { ˆ 𝒙 𝑏 , ˆ 𝒚 𝑏 , ˆ 𝒛 𝑏 } define the (effective)principal axis of the biaxial NS, with ˆ 𝝎 precessing around ˆ 𝒛 𝑏 according to ˆ 𝝎 ( 𝑡 ) = sin 𝜃 𝜔 cos 𝜑 𝜔 ˆ 𝒙 𝑏 + sin 𝜃 𝜔 sin 𝜑 𝜔 ˆ 𝒚 𝑏 + cos 𝜃 𝜔 ˆ 𝒛 𝑏 , (8)where 𝜑 𝜔 ( 𝑡 ) = Ω prec 𝑡 + 𝜑 𝜔 ( ) is the precession phase, with Ω prec the precession frequency (see Zanazzi & Lai 2015, 2020for details). The vector ˆ 𝒙 𝑏 lies in the direction of the projec-tion of ˆ 𝒑 onto the plane perpendicular to ˆ 𝒛 𝑏 .When the precession frequency is much smaller than therotational frequency ( Ω prec (cid:28) 𝜔 ), ˆ 𝝎 remains approximatelyconstant as ˆ 𝒏 rotates around ˆ 𝝎 over the rotational period ofthe NS (in the body frame). Hence ˆ 𝒏 ( 𝑡 ) is described by anequation similar to (7), except in a frame where ˆ 𝒛 𝑏 ≠ ˆ 𝝎 : ˆ 𝒏 ( 𝑡 ) = sin 𝜈 cos 𝜑 𝑛 sin 𝜃 𝜔 ( ˆ 𝒛 𝑏 × ˆ 𝝎 ) × ˆ 𝝎 − sin 𝜈 sin 𝜑 𝑛 sin 𝜃 𝜔 ( ˆ 𝒛 𝑏 × ˆ 𝝎 ) + cos 𝜈 ˆ 𝝎 , (9)where 𝜑 𝑛 ( 𝑡 ) = 𝜔𝑡 + 𝜑 𝑛 ( ) is the spin phase. 3.3. Magnetar undergoing Forced Precession
The right panel of Figure 4 displays our model for forcedprecession, either by a companion (Yang & Zou 2020) or afallback disk (Tong et al. 2020). We work in an inertial ref-erence frame { ˆ 𝒙 𝑖 , ˆ 𝒚 𝑖 , ˆ 𝒛 𝑖 } , with the orbital angular momentumaxis of the companion or disk defining ˆ 𝒛 𝑖 , with ˆ 𝒙 𝑖 lying in thedirection of the projection of ˆ 𝒏 onto the plane perpendicularto ˆ 𝒛 𝑖 . Here, the magnetic longitude 𝛼 is defined as the anglebetween the planes spanned by the vector pairs { ˆ 𝒑 , ˆ 𝝎 } and { ˆ 𝒑 , ˆ 𝒎 𝑓 } . All other quantities are the same as those illustratedin Figure 1.Assuming the spin angular momentum of the NS 𝑳 (cid:39) 𝐼 𝝎 ,with 𝐼 the NS moment of inertia, the NS spin evolves in theforced precession theories according to an equation of theform d ˆ 𝝎 / d 𝑡 | 𝑖 − Ω prec ˆ 𝒛 𝑖 × ˆ 𝝎 =
0, hence has a solution ˆ 𝝎 ( 𝑡 ) = sin 𝜃 𝜔 cos 𝜑 𝜔 ˆ 𝒙 𝑖 + sin 𝜃 𝜔 sin 𝜑 𝜔 ˆ 𝒚 𝑖 + cos 𝜃 𝜔 ˆ 𝒛 𝑖 , (10)where 𝜑 𝜔 ( 𝑡 ) = Ω prec 𝑡 + 𝜑 𝜔 ( ) is the precession phase. Be-cause ˆ 𝒑 and ˆ 𝒎 𝑓 are anchored in the rotating NS, they rotatein the inertial frame according to d ˆ 𝒑 / d 𝑡 | 𝑖 − 𝝎 × ˆ 𝒑 = ˆ 𝒎 𝑓 / d 𝑡 | 𝑖 − 𝝎 × ˆ 𝒎 𝑓 =
0. Since 𝜔 (cid:29) Ω prec , ˆ 𝝎 stays approxi-mately constant as ˆ 𝒑 and ˆ 𝒎 𝑓 rapidly rotate around ˆ 𝝎 , hencethe motion of ˆ 𝒑 ( 𝑡 ) and ˆ 𝒎 𝑓 ( 𝑡 ) are approximately describedby ˆ 𝒑 ( 𝑡 ) = sin 𝜒 cos 𝜑 𝑛 sin 𝜃 𝜔 ( ˆ 𝒛 𝑖 × ˆ 𝝎 ) × ˆ 𝝎 + sin 𝜒 sin 𝜑 𝑛 sin 𝜃 𝜔 ( ˆ 𝒛 𝑖 × ˆ 𝝎 ) + cos 𝜒 ˆ 𝝎 , (11) ˆ 𝒎 𝑓 ( 𝑡 ) = sin 𝜒 𝑓 cos 𝜑 𝑛 sin 𝜃 𝜔 ( ˆ 𝒛 𝑖 × ˆ 𝝎 ) × ˆ 𝝎 + sin 𝜒 𝑓 sin 𝜑 𝑛 sin 𝜃 𝜔 ( ˆ 𝒛 𝑖 × ˆ 𝝎 ) + cos 𝜒 𝑓 ˆ 𝝎 , (12) Li & Zanazziwhere 𝜑 𝑛 ( 𝑡 ) = 𝜔𝑡 + 𝜑 𝑛 ( ) is the spin phase. APPLICATION OF PERIODIC EMISSION MODELTO FRB 180916Now that we have developed a phenomenological emissionmodel for periodic FRBs due to the motion of NSs, we applythese results to the recent multi-wavelength and polarizationmeasurements of FRB 180916. For a dynamical model toremain a plausible explanation for the periodicity of FRB180916, it must explain the following three features of theFRB 180916 emission (Pleunis et al. 2020; Pastor-Marazuelaet al. 2020):1. The linear polarization angle PA varies by less than Δ PA (cid:46) ◦ -20 ◦ for single bursts, and Δ PA (cid:46) ◦ between bursts.2. The bust activity window (region in phase busts seenafter folding over 16.3 day periodicity) widens with adecrease in frequency.3. The burst activity windows in different frequency bandshave phase centers which are offset from one another,or busts at one frequency are systematically delayedwith respect to burst at another frequency. We will callthis effect activity window phase drift with frequency.The first constraint (with Δ PA) lied at odds with the orig-inal predictions of the Zanazzi & Lai (2020) free precessionmodel. This is because the emission region was assumed toemit in the direction of the NS dipole moment ( ˆ 𝒎 𝑓 = ˆ 𝒑 , seeFig. 1). Because FRBs occur when ˆ 𝒏 ≈ ˆ 𝒎 𝑓 = ˆ 𝒑 , the rota-tional phase Ψ (cid:28) (cid:39) sin 𝜒 sin ( 𝜒 − 𝜈 ) Ψ . (13)Since ˆ 𝒏 ≈ ˆ 𝒑 implies 𝜒 ≈ 𝜈 (see Fig. 1), the denomina-tor of equation (13) becomes large, and hence the model ofZanazzi & Lai (2020) predicted large variations in PA duringindividual FRB bursts, as well as modulation of the PA vari-ation between bursts as the NS precessed. However, because ˆ 𝒎 𝑓 and ˆ 𝒑 can have significant differences in orientation (see§2.1), it is possible for PA variations to be sufficiently small tobe consistent with the polarization measurements from FRB180916.In this section, we calculate the variations in PA, as wellas how the burst activity window depends on frequency 𝑓 ,with different dynamical models (Fig. 4). We create maps ofthe range in PA variation, and estimate the activity window ofdifferent models, by first fixing parameters which are constantfor a given dynamical model (see Fig. 1), and in particularspecifying ˆ 𝒎 𝑓 and 𝜉 𝑓 for different frequency bands. Noticehere, we assume the shift of emission region in the observedpolar angle 𝛿 ob while the center of the emission region has the same magnetic longitude 𝛼 . A burst is considered observedat a specific 𝑓 -band if ˆ 𝒏 and ˆ 𝒎 𝑓 lie within an angle 𝜉 𝑓 ofeach other, with no emission if this condition is not met. ThePhase = ( 𝜑 𝑛 + Φ )/( 𝜋 ) for the slow magnetar model, Phase = ( 𝜑 𝜔 + Φ )/( 𝜋 ) for the free/forced precession model (with Φ a constant picked so the 400-800 MHz 𝑓 -band is centeredat Phase = . = 𝜑 𝑛 /( 𝜋 ) are then cycledthrough their possible parameter values ( 𝜑 𝜔 , 𝜑 𝑛 ∈ [ , 𝜋 ] ,middle panels of Fig. 5) to calculate the range of PA valueswith (spin or precession) Phase (top panels of Fig. 5). Thenumber of bursts over all spin phases are then binned byprecessional phase to construct burst activity window profiles(bottom panels of Fig. 5).The top panels of Figure 5 display the PA variation witheither spin (left panels) or precession (middle & right panels)phase. From this, we see all three models are able to havePA variations consistent with observations, as long as theemission region direction ˆ 𝒎 𝑓 has a significant offset fromthe dipole axis ˆ 𝒑 ( 𝛿 obs sufficiently large). The spin phase hassimilar variations as the PA with spin or precession phase,with differences due to the slight differences in geometrybetween the two angles (see eq. [6]).The bottom panels of Figure 5 display the activity windowover different 𝑓 -bands. The magnetar with a slow rotationperiod, as well as the magnetar undergoing free precession,can accommodate an activity window which widens at lower 𝑓 values, as well as activity window phase drifts between 𝑓 -bands. The widening of the activity window is primarily dueto the increase of 𝜉 𝑓 at lower 𝑓 -bands, while the phase driftis primarly due to the higher magnetic polar angle 𝛿 obs val-ues at lower 𝑓 -bands (see §2.1 for discussion). The roughlymonochromatic bust rate with spin phase for the slow magne-tar model is because the event rate is assumed to be uniformover the NS spin phase: dropping this assumption can leadto activity windows which are not monochromatic. We con-clude that the dynamical model of a magnetar with a slowrotation period, as well as a magnetar undergoing free pre-cession, are capable of causing periodic emission consistentwith observations of FRB 180916.The bottom right panel of Figure 5 displays the activity win-dows for the model describing a magnetar undergoing forcedprecession. Although these parameters clearly show a widen-ing activity window with a lower 𝑓 -band, the histogram alsodisplays no activity window phase shift with frequency. Thelack of activity window phase shift is not unique to these par-ticular model parameters, but is rather a general feature of theforced precession model. Consider a magnetar undergoingforced precession, with LOS direction ˆ 𝒏 = sin 𝜃 𝑛 ˆ 𝒙 𝑖 + cos 𝜃 𝑛 ˆ 𝒛 𝑖 .Because ˆ 𝒏 · ˆ 𝒎 𝑓 ( 𝜑 𝜔 , 𝜑 𝑛 ) = ˆ 𝒏 · ˆ 𝒎 𝑓 (− 𝜑 𝜔 , − 𝜑 𝑛 ) (see eqs. [10]& [12]), the activity window is always symmetric about 𝜑 𝜔 = 𝜑 𝑛 =
0, irrespective of the emission frequency 𝑓 .Hence, the observed activity window phase drift with 𝑓 ineriodic FRB Emission from Magnetar Motion 7 P A ( d e g r ee s ) Slow Magnetar . . . . . Sp i n P h a s e . . . . . . Phase (Spin) N u m b e r o f B u r s t s ( a r b ) − P A ( d e g r ee s ) Free Precession . . . . . . . Sp i n P h a s e . . . . . . Phase (Precession) N u m b e r o f B u r s t s − . − . − . − . . . . P A ( d e g r ee s ) Forced Precession . . . . . . Sp i n P h a s e . . . . . . Phase (Precession) N u m b e r o f B u r s t s ( a r b ) Figure 5.
Predicted properties of three different models for periodic FRBs: position angle PA (top panels), Spin Phase = 𝜑 𝑛 /( 𝜋 ) (middlepanels), and activity windows (bottom panels, Number of Bursts normalization arbitrary), for the frequency bands 1.36-2 GHz (green), 400-800MHz (orange), and 110-180 MHz (blue). The observed properties are folded over the timescale which causes the 16.3 day periodicity, whichis the rotation period (Phase = [ 𝜑 𝑛 + Φ ]/[ 𝜋 ] , left panels), or the precession period (Phase = [ 𝜑 𝜔 + Φ ]/[ 𝜋 ] , middle & right panels), with Φ chosen so that the CHIME band (400-800 MHz) activity window is centered at Phase = . Left panels:
Emission from a magnetar withslow rotation (§3.1). Note different frequency bands overlap in PA and Spin Phase. Fixed model parameters are 𝛿 obs = ◦ , 45 ◦ , 70 ◦ , and 𝜉 𝑓 = ◦ , ◦ , 20 ◦ , for frequency bands 1.36-2 GHz, 400-800 MHz, 110-180 MHz, respectively, with 𝛼 = ◦ , 𝜈 = ◦ , and 𝜃 𝑝 = ◦ . Middlepanels:
Emission from a magnetar undergoing free precession (§3.2). Fixed model parameters are 𝛿 obs = ◦ , ◦ , ◦ , and 𝜉 𝑓 = ◦ , ◦ , ◦ ,for frequency bands 1.36-2 GHz, 400-800 MHz, 110-180 MHz, repectively, with 𝛼 = ◦ , 𝜈 = ◦ , 𝜃 𝜔 = ◦ , and 𝜃 𝑝 = ◦ . Right panels:
Emission from a magnetar undergoing forced precession (§3.3). Fixed model parameters are 𝛿 obs = ◦ , ◦ , ◦ , and 𝜉 𝑓 = ◦ , ◦ , ◦ , forfrequency bands 1.36-2 GHz, 400-800 MHz, 110-180 MHz, respectively, with 𝛼 = ◦ , 𝜒 = ◦ , 𝜃 𝜔 = ◦ , and 𝜃 𝑛 = ◦ . FRB 180916 disfavors the dynamical model of a magnetarundergoing forced precession. PROPOSED OBSERVATIONAL TESTSThe previous section showed the dynamical model of amagnetar with a slow rotation period, and a magnetar un-dergoing free precession, could explain the polarization andfrequency-dependent activity window of FRB 180916, whilethe magnetar undergoing forced precession was disfavored.In this section, we describe how further polarization and ac-tivity window measurements can refine constraints on modelparameters, and additional observations can favor or rule-outthe remaining dynamical theories to explain periodic FRBs.If additional periodic FRB sources are detected, the PAswings and activity windows are expected to change fromsource to source. Within the context of our phenomenological emission model, the phase-drift of the FRB activity windowwith frequency results from a change of emission region.This phase-drift can be in either direction, depending on thedirection of rotation/precession, as well as the orientationof the emission regions with respect to each-other. Eventrates of FRBs can also differ significantly across frequencydue to different magnetic field geometries. However, boththe slowly-rotating and freely-precessing magnetars requirethe emission regions to be asymmetric with respect to theNS dipole axis. In contrast, it is difficult for the inducedprecession model to have an activity window which driftsasymmetrically in phase with frequency.All dynamical models allow a small PA change with pre-cession or spin phase, while larger PA changes like in thecase of FRB 180301(Luo et al. 2020) is also allowed. ThePA variations of both models differ between frequency bands. Li & ZanazziHowever, for a slowly-rotating magnetar, there is a constantPA angle at each (spin) phase, while for the freely-precessingmagnetar, a range of PA values are expected at each (pre-cession) phase (Fig. 5, upper panels). Moreover, for a fixedprecession phase, only certain spin phases will lead to anobserved burst (Fig. 5, middle panels). More accurate PAmeasurements from high S/N bursts would be able to distin-guish between these two models (or rule out both).The precession model also requires fixed emission region,and a small timescale periodicity due to the rotation periodof the magnetar. This short periodicity may be concealed byvarious timing noises common in young magnetars. How-ever, if a short period is found, the precession model predictsa change of the short periodicity duty cycle with the NS pre-cession phase. For instance, in Figure 5 central middle panel,the spin duty cycle appears smaller at the edge of each ac-tive (precession) phase. Moreover, the active window in thespin phase is changing against frequency with this setup ofasymmetric emission against magnetic pole.To get the same activity windows, the free-precessionmodel requires a smaller emission region than the slowly-rotating magnetar model. This is because for each spin phaseof the slowly-rotating magnetar, the LOS points at a specificlocation on the NS surface, while for each precession phaseof the freely-precessing magnetar, the LOS rotates around theNS rotation axis. Distinguishing between these two modelswould constrain the FRB emission region altitude and angularsize.Recently, Katz (2020) discussed how detecting a change inthe FRB period can further differentiate between dynamicaltheories. We note the freely-precessing magnetar model ex-pects a much larger period change than the slowly-rotatingmagnetar model, since the spin-down timescale is much shorter in the former model due to the faster rotation fre-quency (e.g. Zanazzi & Lai 2020). SUMMARY AND CONCLUSIONSIn this paper, we construct a phenomenological model ofFast Radio Burst (FRB) emission from rotating magnetars totest periodic FRB theories, in light of recent measurementsof small polarization Position Angle (PA) swings during andbetween bursts, as well as the narrowing and phase shift ofburst activity windows with frequency. Our model assumesburst are emitted from a region anchored into the magnetarand offset from the dipole axis, with region size and dipoleoffset angle increasing with a decrease in frequency, due tothe altitude dependence of curvature radiation on frequency(§2.1). The model PA values are given by the rotating vectormodel (§2.2).Using this model, we constrain three separate dynamicalmodels which have been invoked to explain the 16.3 dayperiodicity of FRB 180916: a magnetar with a slow rota-tion period (§3.1), a magnetar undergoing free precession(§3.2), and a magnetar undergoing forced precession due toan external body (§3.3). We find the slowly-rotating andfreely-precessing magnetar models can produce PA swingsand frequency-dependent activity windows consistent withrecent observations, but the magnetar undergoing forced pre-cession is disfavored as a dynamical model, due to its inabilityto produce a frequency-dependent phase drift of the burst ac-tivity window (§4). Future observations are necessary todistinguish between the remaining theories, and understandwhat is causing the periodicity of FRB 180916 (§6)ACKNOWLEDGEMENTSDL acknowledges the discussion with Liam Connor andVikram Ravi. JZ is supported by a CITA postdoctoral fellow-ship.REFERENCES
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